Volume 33, Issue 9, September 1992
Index of content:

Localized coherent structures of the Davey–Stewartson equation in the bilinear formalism
View Description Hide DescriptionThe Davey–Stewartson equation is considered from the point of view of the bilinear formalism of the Kyoto school. Multidromion solutions are constructed in terms of free fermions and their asymptotic properties are characterized. The dynamical properties of dromions are discussed.

WKB equivalent potentials for q‐deformed harmonic and anharmonic oscillators
View Description Hide DescriptionWKB equivalent potentials (WKB‐EP’s) giving the same WKB spectrum as the q‐deformed harmonic oscillator are determined in analytic form. For q being complex the WKB‐EP resembles the Pöschl–Teller, Morse, or Woods–Saxon potentials, widely used in nuclear, hypernuclear, and molecular physics. For q real, the WKB‐EP still goes to infinity at large x, while at small x it resembles a sextic oscillator. The WKB‐EP’s of the anharmonic oscillators with U_{ q }(2) or SU_{ q }(1,1) symmetries, used for the description of vibrational molecular spectra, are also calculated. They correspond to q‐deformed versions of the modified Pöschl–Teller potential.

Symmetry algebras of third‐order ordinary differential equations
View Description Hide DescriptionThe main result of this paper is the complete classification of the third‐order ordinary differential equations according to their symmetries. The same classification was done for second order by Tresse [‘‘Détermination des invariants ponctuels de l’équation différentielle ordinaire du second ordre y‘=ω(x,y,y’),’’ Gekrönte Preisschrift, Hirzel, Leipzig (1896)], and recently for arbitrary order linear ordinary differential equations. The sections preceding the classification consist of a brief description of the concepts and methods along the lines of Krause and Michel [Lecture Notes Phys. 382, 251 (1991)]. These sections also contain some definitions and a table listing the prolongations of a few vector fields. Finally, two appendices give additional information relevant to equations in real variables and describe how some of the results can be easily generalized to higher orders.

A new boson realization for the isotropic three‐dimensional oscillator
View Description Hide DescriptionTwo new closely related boson realizations of the Heisenberg–Weyl algebra are derived that are directly equivalent to the introduction of spherical coordinates for the isotropic three‐dimensional harmonic oscillator. This provides another way to obtain matrix elements of all physical operators connecting angular‐momentum eigenstates, which are represented here by simple products of boson operators acting on the vacuum. Two of these bosons carry the angular‐momentum quantum numbers while the third is a novel ‘‘radial boson.’’ In this approach, one begins by finding the nonunitary Dyson and then the unitarized Holstein–Primakoff realizations for the spectrum‐generating SU(1,1)×SO(3) algebra of the oscillator. The corresponding mappings of the Heisenberg–Weyl generators are then obtained from their tensor properties under SU(1,1)×SO(3). The Dyson map of the Heisenberg–Weyl generators is not uniquely determined in this way, but the Holstein–Primakoff map is nevertheless unique. The possibility exists for generalization to isotropic oscillators of higher dimension, which are of interest in connection with nuclear collective motion.

Addition formulas for q‐Bessel functions
View Description Hide DescriptionThe two‐dimensional Euclidean algebra provides an algebraic setting for q‐Bessel functions. Using a two‐variable realization of this algebra, q analogs of the Lommel and Graf addition formulas for the q‐Bessel functions are obtained.

Curvature inheritance symmetry in Riemannian spaces with applications to fluid space times
View Description Hide DescriptionKatzin et al. [G. H. Katzin, J. Levine, and W. R. Davis, J. Math. Phys. 10, 617 (1969)] introduced curvature collineations (CC), defined by a vector ξ, satisfying L _{ξ} R _{ bcd } ^{ a }=0, where R _{ bcd } ^{ a } is the Riemann curvature tensor of a Riemannian space V _{ n } and L _{ξ} denotes the Lie derivative. They proved that a CC is related to a special conformal motion which implies the existence of a covariant constant vector field. Unfortunately, recent study indicates that the existence of a covariant constant vector restricts V _{ n } to a very rare special case with limited physical use. In particular, for a fluid space time with special conformal motion, either stiff or unphysical equations of state are singled out. Moreover, perfect fluid space times do not admit special conformal motions. This information was not available, in 1969, when CC symmetry was introduced. In this paper, CC is generalized to another symmetry called ‘‘curvature inheritance’’ (CI) satisfying L _{ξ} R _{ bcd } ^{ a }=2αR _{ bcd } ^{ a }, where α is a scalar function. We prove that a proper CI (i.e., α≠0) has direct interplay with the physically significant proper conformal motions. As an application, we show that a proper CI, which is also a conformal Killing vector (CKV), can generate new and physically relevant solutions for a variety of fluid spacetimes. In particular, it is shown, that, for CI with CKV, the known stiff or unphysical equations of state are not singled out.

Reflection principles for biased correlated walks. Simple applications
View Description Hide DescriptionThe new type of reflection principle applicable to the Laplace transform of the arrival probability for biased walks developed by Khantha and Balakrishnan is extended to the cases of biased correlated walks with absorbing and/or reflecting walls. An exact expression for the mean escape time <t≳ for the linear discrete walk bounded by absorbing walls at 0 and N is obtained as follows: <t≳=N(1−δ)[2(N−1)(a−d)]^{−1}[(a ^{ N }+d ^{ N })/ (a ^{ N }−d ^{ N })−(1+δ)/(a−d)] +δ[(N−1)(a−d)]^{−1}{1−[N(a−d)d ^{ N−1}]/ (a ^{ N }−d ^{ N })}, where a=1/2(a+δ+ε) and d=1/2(1+δ−ε), and δ and ε are the degrees of correlation and anisotropy, respectively.

Canonical perturbation expansions to large order from classical hypervirial and Hellmann–Feynman theorems
View Description Hide DescriptionThe classical hypervirial and Hellmann–Feynman theorems are used to formulate a ‘‘perturbation theory without Fourier series’’ that can be used to generate canonical series expansions for the energies of perturbed periodic orbits for separable classical Hamiltonians. As in the case where these theorems are used to generate quantum mechanical Rayleigh–Schrödinger perturbation series, the method is very efficient and may be used to generate expansions to large order either numerically or in algebraic form. Here, the method is applied to one‐dimensional anharmonic oscillators and radial Kepler problems. In all cases, the classical series for energies and expectation values are seen to correspond to the expansions associated with their quantum mechanical counterparts through an appropriate action preserving classical limit as discussed by Turchetti, Graffi, and Paul. This ‘‘action fixing’’ is inherent in the classical Hellmann–Feynman theorem applied to periodic orbits.

Integrability of seventh‐order KdV‐like evolution equations using formal symmetries and perturbations of conserved densities
View Description Hide DescriptionSeventh‐order evolution equations of the form u _{ t }=d/dx where G is a differential polynomial with certain scaling properties are classified using the integrability test of Mikhailov–Shabat–Sokolov based on the existence of formal symmetries. The integrability of these equations is checked using Kodama’s method based on the perturbation of the conserved densities of the KdV equation. According to these tests, the only integrable KdV‐like seventh‐order equations are the KdV, Sawada–Kotera, and Kaup equations. It is also shown that these equations are characterized by the existence of two conserved densities.

Pulse formation in a dissipative nonlinear system
View Description Hide DescriptionA nonlocal nonlinear evolution equation is proposed that describes pulse formation in a dissipative system. A novel feature of the equation is that it can be solved exactly through a linearization procedure. The solutions are constructed under appropriate initial and boundary conditions and their properties are investigated in detail. Of particular interest is pulse formation, which is caused by a balance between nonlinearity and dissipation. The asymptotic behavior of the solution for large time is then represented by a train of moving pulses with equal amplitudes. The corresponding position of each pulse is shown to be characterized by the zero of the Hermite polynomial, irrespective of initial conditions.

Scalar and spinor solutions in the space‐time of a domain wall
View Description Hide DescriptionThe Klein–Gordon and Weyl equations are solved in the space‐time of a plane vacuum domain wall. The solutions depend on global features of this space‐time. In the spinor case the current is computed, showing its dependence on global features as well.

Bernstein processes and spin‐1/2 particles
View Description Hide DescriptionA pure jump Bernstein process whose probability density is the product of the solution of the (imaginary time) Schrödinger equation and its adjoint equation, associated to a class of time dependent Hamiltonians describing spin‐1/2 particles, is constructed. A path integral representation of these solutions, in terms of this process, is obtained as well as a calculus over path space for the associated Euclidean quantum observables.

The weak coupling limit for quadratic interaction
View Description Hide DescriptionThe weak coupling limit of a ‘‘system+reservoir’’ model with nonlinear (quadratic) interaction is discussed. In a previous paper [Accardi and Lu, Stoch. Process. Phys. Geom., 1–26], it has been shown that, if the matrix elements of the van Hove rescaled evolution are taken with respect to the same collective vectors used for linear interactions, then the limiting evolution is not unitary. In the present paper, we choose a new type of collective vectors and obtain, in the weak coupling limit, a unitary evolution satisfying a quantum stochastic differential equation. A remarkable feature of the limiting process is that the quadratic nature of the interaction shows up in a nontrivial choice of the Hilbert space where the limiting quantum noise takes its values, which is qualitatively new with respect to all the models considered up to now.

Symmetry and quantization: Higher‐order polarization and anomalies
View Description Hide DescriptionThe concept of (full) polarization subalgebra in a Group Approach to Quantization on a Lie groupG̃ as a generalization of the analogous concept in geometric or standard quantization is discussed. The lack of full polarization subalgebras is considered as an anomaly of the corresponding system and related to its more conventional definition. A generalization of the subalgebra of (full) polarization is then provided, made out of higher‐order differential operators in the enveloping algebra of G̃. Higher‐order polarizations can also be used to quantize nonanomalous theories in different ‘‘representations.’’ Numerous examples are analyzed, including the finite‐dimensional dynamics associated with the Schrödinger group, which presents an anomaly, and an infinite‐dimensional anomalous system associated with the Virasoro group. In the last example, the operators in the higher‐order polarization are in one‐to‐one correspondence with the null vectors in the Verma module approach.

Uniqueness of the scalar product in the tensor product of quaternion Hilbert modules
View Description Hide DescriptionIn the construction of a tensor product of quaternion Hilbert modules, given in a previous work (real, complex, and quaternionic), inner products were defined in the vector spaces formed from the tensor product of quaternion algebrasH modulo an appropriate left ideal in each case. Under conditions that are necessary for the definition of a scalar product in the quaternionic Hilbert modules and a natural condition on the algebraic structure, it is proven that the scalar products which are defined are unique.

On the Coulomb singular kernel of Lippmann–Schwinger‐type equation
View Description Hide DescriptionA Lippmann–Schwinger‐type equation with Coulomb singular kernel is considered. It is shown that all its solutions are singular on the energy shell, k ^{2}/2μ=E. In order for this equation to have a solution with a singularity of the type (E−k ^{2}/2μ+iε)^{ iα}, α is shown to be equal to the Coulomb parameter η. The Coulomb singular kernel in the given class of functions is found to split into a δ function and a kernel which smoothes the singularity.

Vacuum energy for 3+1 dimensional space‐time with compact hyperbolic spatial part
View Description Hide DescriptionThe vacuum energy for a massless scalar field defined on a hyperbolic manifoldH ^{3}/Γ is evaluated using zeta‐function and heat kernel regularization techniques and Selberg trace formula for cocompact group Γ. A negative contribution to the vacuum energy associated with trivial line bundle related to character χ=1 is found.

Algebraic structures of the quantum projective [sl(2,C)‐invariant] field theory: The commutative version of Elie Cartan’s exterior differential calculus
View Description Hide DescriptionThe commutative exterior differential calculus, appearing in the quantum projective field theory, is described.

D‐dimensional torus as compact manifold and Kaluza–Klein cosmological model
View Description Hide DescriptionSingularity‐free solutions of higher‐dimensional Einstein field equations are obtained in the background of M ^{4}×T ^{ D } manifold (M ^{4} is a usual four‐dimensional Friedmann– Robertson–Walker model and T ^{ D } is a D‐dimensional torus). Moreover, through dimensional reduction and one‐loop quantum correction to scalar field, time‐dependent cosmological constant Λ, effective gravitational constant G _{eff}, and a fine‐structure constant e/4π are derived in the effective four‐dimensional theory using solutions of Einstein’s equations. It is found that at late times Λ≂0.

Construction of a complex‐valued fractional Brownian motion of order N
View Description Hide DescriptionIn this paper, a Brownian motion of order n (n≳2) is defined by a probabilistic approach different from Hochberg’s and Mandelbrot’s. This process is constructed from sums of independent R _{+} ^{1/n }‐valued random variables (rv) (where R _{+} ^{1/n }={z∈C; z ^{ n }∈R _{+}}). Many properties of the real standard Brownian motion are generalized at order n, but in the case n≳2, it is interesting to describe the Brownian motion of order n on the σ algebra ⊗[B(R _{+} ^{1/n })] ^{ R + } [where B(R _{+} ^{1/n }) is the σ algebra generated by sets of type A(0,h)={z∈C; z ^{ n }∈[0;h ^{ n }[},(hεR _{+} ^{*})]. This σ algebra is totally different from ⊗[B(R)] ^{ R + }. Thus this study shows the fractal nature of the Brownian motion of order n, and given invariance scale (self‐similarity) properties. Then, a stochastic integral and an Itô–Taylor lemma at order n are given to allow the representation of the solution of the heat equation of order n by a probabilistic average. All these results can be obtained via nonstandard analysis methods (infinitesimal time discretization). Finally, one remarks that this process has a.s (almost surely) continuous sample paths, infinite variance, and independent increments, whereas the fractional Brownian motion of Mandelbrot has a.s continuous sample paths, finite variance, and interdependent increments.